Optimal. Leaf size=53 \[ \frac {x}{3 d^2 \sqrt {d^2-e^2 x^2}}+\frac {2 (d+e x)}{3 e \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {653, 191} \begin {gather*} \frac {x}{3 d^2 \sqrt {d^2-e^2 x^2}}+\frac {2 (d+e x)}{3 e \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 191
Rule 653
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\frac {2 (d+e x)}{3 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{3} \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac {2 (d+e x)}{3 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x}{3 d^2 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 47, normalized size = 0.89 \begin {gather*} \frac {(2 d-e x) (d+e x)}{3 d^2 e (d-e x) \sqrt {d^2-e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.37, size = 42, normalized size = 0.79 \begin {gather*} \frac {(2 d-e x) \sqrt {d^2-e^2 x^2}}{3 d^2 e (d-e x)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 71, normalized size = 1.34 \begin {gather*} \frac {2 \, e^{2} x^{2} - 4 \, d e x + 2 \, d^{2} - \sqrt {-e^{2} x^{2} + d^{2}} {\left (e x - 2 \, d\right )}}{3 \, {\left (d^{2} e^{3} x^{2} - 2 \, d^{3} e^{2} x + d^{4} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 48, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {-x^{2} e^{2} + d^{2}} {\left (x {\left (\frac {x^{2} e^{2}}{d^{2}} - 3\right )} - 2 \, d e^{\left (-1\right )}\right )}}{3 \, {\left (x^{2} e^{2} - d^{2}\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 44, normalized size = 0.83 \begin {gather*} \frac {\left (e x +d \right )^{3} \left (-e x +d \right ) \left (-e x +2 d \right )}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{2} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 58, normalized size = 1.09 \begin {gather*} \frac {2 \, x}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, d}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e} + \frac {x}{3 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.48, size = 38, normalized size = 0.72 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (2\,d-e\,x\right )}{3\,d^2\,e\,{\left (d-e\,x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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